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    Lunar distance method of Chauvenet
    From: Paul Hirose
    Date: 2020 May 25, 21:59 -0700

    In another thread I mentioned the lunar distance method of William
    Chauvenet. I think it first appeared in 1851, in the Astronomical Journal:
    
    
    http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1851AJ......2...24C&db_key=AST&page_ind=0&data_type=GIF&type=SCREEN_VIEW&classic=YES
    
    The American Ephemeris and Nautical Almanac for the Year 1855 contains a
    description in "navigator language," two examples, and tables:
    
    https://archive.org/details/americanephemer19offigoog/page/n504/mode/2up
    
    His manual of spherical astronomy (vol. 1) says it's "the shortest and
    simplest of the approximative methods when these are rendered
    sufficiently accurate by the introduction of all necessary corrections
    ... There are briefer methods to be found in every work on Navigation,
    which will (and should) be preferred in cases where only a rude
    approximation to the longitude is required.":
    
    https://archive.org/details/manualofspherica01chauiala/page/402/mode/2up
    
    The tables are in volume 2:
    
    https://archive.org/details/manualofspherica02chau/page/600/mode/2up
    
    
    Chauvenet gives two examples in the 1855 almanac. Since my objective was
    to evaluate his "approximate method" or more precisely, my computer
    implementation, I calculated new simulated observations with Lunar 4.4.
    The angles are within a few arc minutes of the original examples.
    
    I could have used those examples unchanged, but that would introduce
    errors (such as the lunar ephemeris) which are not the fault of the
    Chauvenet method itself.
    
    The first is a morning Sun lunar. (Of course in practice the correct
    time and longitude would not be known.)
    
    1855-09-07 08:07 UT1 (correct time)
    +10 s delta T
    35.5° N 30° W (correct position)
    zero height of eye
    75 F (24 C), air pressure 29.1 inches Hg (985 mb)
    
    observations:
    
    49°25.66' Moon lower limb refracted altitude
    5°18.87' Sun lower limb refracted altitude
    43°52.71' observed lunar distance, near to near
    
    Almanac data:
    
    Moon at 8 h UT1
    14.78' geocentric semidiameter
    54.27' horizontal parallax
    +25°18.01' declination
    
    Sun at 8 h UT1
    15.88' semidiameter
    0.15' horizontal parallax
    +6°15.61' declination
    
    geocentric apparent distances (center to center)
    08 h UT1 45°08.52'
    09 h UT1 44°41.31'
    10 h UT1 44°14.12'
    
    (Back in the day, lunar distances were typically tabulated every three
    hours. However, Chauvenet's method doesn't require that. My
    implementation accepts distances at any time interval. Also, it applies
    a correction for second differences, i.e., a uniform lunar distance rate
    is not assumed. The method of second difference correction is explained
    in the interpolation section of his spherical astronomy book.)
    
    Chauvenet solution:
    
    45°05.300′ geocentric distance
    8h07m06s observation time
    
    correct values:
    45°05.34' distance
    8m07m00s time
    
    Chauvenet solution error = +6 s.
    
    
    The other example in the 1855 almanac is an evening Fomalhaut lunar.
    
    1855-08-30 05:40 UT1 (correct time)
    +10 s delta T
    55° 20' S 120° 25' W (correct position)
    zero height of eye
    20 F (-7 C), 31 inches Hg (1050 mb) altimeter setting
    
    observations
    
    6°15.75' Moon lower limb refracted altitude
    52°23.15' Fomalhaut refracted altitude
    46°29.09' Moon far limb to Fomalhaut observed distance
    
    almanac data
    
    Moon at 6 h UT1
    16.40' geocentric semidiameter
    1°00.19' horizontal parallax
    +3°54.45' declination
    
    Fomalhaut
    -30°23.13' declination
    
    geocentric apparent distance from Moon center:
    05 h UT1 44°59.00'
    06 h UT1 45°32.55'
    07 h UT1 46°06.13'
    
    Chauvenet solution:
    
    45°21.352′ geocentric distance
    5h39m59s observation time
    
    45°21.36' correct distance
    5h40m00s correct time
    
    solution error = -1 s
    
    The examples in the 1855 almanac appear to have been constructed to
    flatter Chauvenet's lunar method vs. methods that make no correction for
    nonstandard atmosphere conditions. One example is at high temperature
    and low barometer, the other at low temperature and high barometer. Both
    have one body at low altitude.
    
    
    Finally, there's the Sun lunar in his "Manual of Spherical and Practical
    Astronomy." In this case my re-calculated example attempts to match
    Chauvenet's angles to a tenth minute.
    
    1856-03-10 03:12:49 UT1 (correct time)
    +10 s delta T
    34°56.6' N 149°39.8' W
    zero height of eye
    
    15.6 C (60.0 F) at observer
    999.0 mb (29.50″ Hg) air pressure
    50.0% relative humidity
    
    observations
    
    52°34.0' Moon lower limb altitude
      8°56.4' Sun lower limb altitude
    44°37.0' Moon to Sun, near to near
    
    almanac data at 0300 UT1
    
    Moon
    16.34' geocentric semidiameter
    59.97' horizontal parallax
    +14°15.18' declination
    
    Sun
    16.09' geocentric semidiameter
    0.15' horizontal parallax
    -4°03.32' declination
    
    geocentric apparent distance, center to center
    3 h UT1 45°40.88'
    4 h UT1 46°14.58'
    5 h UT1 46°48.24'
    
    Chauvenet solution
    
    45°48.098′ geocentric distance
    3h12m51s observation time
    
    45°48.08′ correct geocentric distance
    3h12m49s correct time
    
    +2 s time error
    
    Chauvenet's approximate method had good accuracy in all my tests. (I
    have never evaluated his rigorous method.) However, I believe the most
    direct route to maximum accuracy in a computer solution is to seek the
    time and place that duplicates the three observed angles. This allows
    all factors that affect the topocentric angles to be applied, to the
    extent that they can be calculated.
    

       
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